Producer Price Index Manual: Theory and Practice ... - METAC

Producer Price Index Manual
Let us say the price of a product m available in
January (period t) is unavailable in March (period t
+ 2). The price of product m can be predicted for
March by inserting the characteristics of the old
unavailable product m into the estimated regression
equation for March, this process is repeated
for successive months. The predicted price for the
old product in March and the price comparison
with January (period t) are given, respectively, by
(7.25a)
pˆ
= exp ⎡β ˆ + β z ⎤
⎣ ∑ ⎦
,
t+ 2 t+ 2 t+
2 t
m k k k,
m
t+
2 t
and pˆ m
/ pm
, that is, the old model’s price is adjusted.
In the example in Table 7.2(a), pˆ
ˆ
3 4
2,
p
2
, etcetera
and 3 4
ˆp
6
, ˆp
6
, etcetera would be estimated
1
1
and compared with p
2
and p
6
, respectively. The
blanks for products 2 and 6 in Table 7.2(a) would
be effectively filled in by the estimated price from
the regression equation.
7.140 An alternative procedure is to select for
each unavailable m product a replacement product
n. In this case, the price of n in period t + 2 is
known, and a predicted price for n in period t is required.
The predicted price for the new product
and required price comparison are:
(7.25b)
t t t t 2
pˆ
n
= exp ⎡
⎣β 0
+ ∑ βk z + ⎤
k,
m⎦,
t+
2 t
and p / ˆ
n
pn
, that is, the new model’s price is adjusted.
In this case, the characteristics of product n
are inserted into the right-hand side of an estimated
regression for period t. The price comparisons of
equation (7.25a) may be weighted by w t m
, as would
those of its replaced price comparison in equation
(7.25b).
7.141 A final alternative is to take the geometric
mean of the formulations in equations (7.25a) and
(7.25b) on grounds analogous to those discussed in
Chapter 15 and by Diewert (1997) for similar index
number issues.
7.142 Hedonic imputation: predicted vs. predicted—A
further approach is the use of predicted
values for the product in both periods, for example,
t+
2 t
pˆ
/ ˆ
n
pn, where n represents the product. Consider
a misspecification problem in the hedonic equation.
For example, there may be an interaction effect
between a brand dummy and a characteristic,
say between Dell and speed in the example in Table
7.3. Having both characteristics may be worth
more on price (from a semi-logarithmic form) than
their separate individual components (for evidence
of interaction effects see, Curry, Molgan and Silver,
2000). The use of p / ˆ
t+
2 t
n
pn
would be misleading
since the actual price in the numerator would
incorporate the 5 percent premium while the one
predicted from a straightforward semi-logarithmic
form would not. It is stressed that in adopting this
approach, a recorded, actual price is being replaced
by an imputation. Neither this nor the form of bias
discussed above are desirable . Diewert (2002e)
considers a similar problem and suggests an adjustment
to bring the actual price back in line with
the hedonic one.
7.143 The comparisons using predicted values in
both periods are given as
pˆ
pˆ
t+
2
n
t+
2
m
/ pˆ
for the new product,
t
n
/ pˆ
t
m
for the disappearing or old product, or
t+ (7.26) ( 2 t t+
ˆ / ˆ )( ˆ 2 t
⎡ p p p / pˆ
) ⎤
12
⎣ n n m m ⎦
as a (geometric) mean of the two.
7.144 Hedonic adjustments using coefficients—
In this approach, a replacement product is used and
any differences between the characteristics of the
replacement n in t + 2 and m in period t are ascertained.
A predicted price for n in period t, that is,
ˆ t
n
t 2
p is compared with the actual price, p +
n
. However,
unlike the formulation in equation (7.25b) for
example, p ˆ t
n
, may be estimated by applying the
subset of the k characteristics that distinguished m
from n, to their respective implicit prices in period
t estimated from the hedonic regression, and ad-
t
justing the price of p
m
. For example, if the nearest
replacement for product 2 was product 3, then the
characteristics that differentiated product 3 from
product 2 are identified and the price in the base
1
period p3
is estimated by adjusting p 1 2
using the
appropriate coefficients from the hedonic regression
in that month. For example, for washing machines,
if product 2 had an 800 revolutions pre
minute (rpm) spin speed and product 3 had an
1,100 rpm spin speed, other things being equal, the
shadow price of the 300 rpm differential would be
estimated from the hedonic regression, and p
1 2
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